The map x → x x modulo p is related to a variation of the ElGamal digital signature scheme in a similar way to the discrete exponentiation map, but it has received much less study. We explore the number of fixed points of this map by a statistical analysis of experimental data. In particular, the number of fixed points can in many cases be modeled by a binomial distribution. We discuss the many cases where this has been successful, and also the cases where a good model may not yet have been found.
When ordered by discriminant, it is known that about 83% of quartic fields over Q have associated Galois group S 4 , while the remaining 17% have Galois group D 4 . We study these proportions over a general number field F . We find that asymptotically 100% of quadratic number fields have more D 4 extensions than S 4 and that the ratio between the number of D 4 and S 4 quartic extensions is biased arbitrarily in favor of D 4 extensions. Under GRH, we give a lower bound that holds for general number fields.
To any quartic D4 extension of Q, one can associate the Artin conductor of a 2dimensional irreducible representation of the group. Altug, Shankar, Varma, and Wilson determined the asymptotic number of such fields when ordered by conductor. We refine this, realizing a secondary term and power saving error term.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.